Johannah,

Thanks for this insightful presentation and the project on deepening mathematics learning among multilingual learners.

Duagrams are at the core of problem solving and and design, and the kind of work you’ve shared elicits learners higher level cognitive abilities. It is also true that communication is richer when we develop tluency in using text, verbal, diagrammatic and non-verbal approaches. The task of translating word problems into diagrams is quite demanding and I appreciate your project’s intervention from middle school.

i had an experience some years back with secondary students in Kenya, where ability to work out problems on relative motion using diagrams was much lower compared to use of the computational method, interestedly even among top achievers. The other experience is when I was pursuing my studies in Japan, my limited language was supplemented by use of diagrams with limited English text, gestures and use of basic verbs in Japanese. 

I guess you’ve seen the Presentation by Eric Hamilton in the current showcase where there are similarities is enhancement of multiple intelligences in diverse cultures

Thanks for visiting and sharing your experiences, Ateng' Ogwel. We agree that being able to communicate through multiple modalities, including using diagrams, strengthens communication, and it sounds like you had some first-hand experience with this during your studies in Japan.

And yes, learning to use diagrams in problem solving can take time and scaffolding for students. We had success in these lessons with scaffolding diagramming by using partial diagrams for students to complete and by providing worked examples showing diagramming approaches for students to analyze. Teachers and students found these strategies helped get students started on using diagrams. We will check out Eric Hamilton's presentation--thanks for the suggestion.

Thanks for sharing, Laura. Sounds like you’re doing great work in your algebra support classes and we’re so glad to hear you’ve found these types of strategies and lessons useful there.

We definitely agree about the importance/challenge of fraction division! We chose it as a focal point for this study and these lessons knowing that understanding of fraction operations is so important to students’ mathematics mathematical learning (in 6^th grade and beyond), that diagrams are important tools in this content, and that this is very challenging content to teach and learn so resources could have an impact.

We are exploring next steps related to this work. We don’t have immediate plans to develop lessons in other content areas, but would be interested in hearing what you and others would find most useful. Through other work we are also thinking about professional learning resources about the kinds of strategies that were embedded in these lessons. (See for example resources at http://go.edc.org/vam.) If you teach algebra support classes, a curriculum you might be interested in from some of our colleagues at EDC is Transition to Algebra. It was not designed specifically for multilingual learners, but it does rely heavily on various visual representations of mathematics.

To your question about promoting oral communication in mathematics class, including for reluctant speakers: one way we do that in these lessons is through structured pairs work integrating use of sentence starter cards. In each lesson, after students start drawing their own diagrams for a problem, they then meet with a partner to use sentence starters such as “In this diagram, I see…” and “In this diagram, I don’t see…” to talk about each other’s diagrams. During partner work, everyone in the classroom has the chance to talk about the mathematics rather than just one or a few students at a time, and it is a low pressure opportunity to engage in mathematical talk because it is in front of just one other student. Providing every student with a set of sentence starter cards on a ring (e.g., see the handout provided as a resource with this video) also allows students to structure their discussion about the diagrams, and the teacher and/or student volunteers can model how to use the sentence starter cards to talk about a diagram when introducing the cards. We’d love to hear other ideas about how you all promote and support oral communication!

Thanks, Dan! We always appreciate your insights into this work.

We decided to take on fraction division because we thought that through showing the how and why of fraction division to middle grades’ students may help to build their understandings of and appreciation for fraction operations. Through this project, exploring how diagrams can support students in their understanding and exploration of fraction division has been so interesting. In terms of type of diagram, we included a strong focus on number lines to help emphasize fractions as numbers with magnitude as recommended through research by Siegler and colleagues, but also supported students to develop various diagrams in their problem solving, including area models, to facilitate their own sense-making and attention to relationships in fraction division contexts.

To promote and scaffold diagram use, we incorporated strategies such as providing partial diagrams for students to complete and worked examples that show diagramming approaches for students to analyze and discuss. During our initial analyses of interviews and written reflections, we have seen a variety of student diagrams and students have shared how number lines help them to organize their thinking and begin solving fraction division tasks.  Teachers also shared that they have found that integrating diagrams into their instruction has helped to them explain more about the relationships in fraction division and focus on what the whole unit is in each case; for example, in a diagram, teachers shared that they could emphasize a third of what or a fourth of a different quantity more easily than when just a calculation or algorithm was used.  The partial diagrams and worked examples were also scaffolds that teachers found useful for getting students started with creating diagrams.

Thanks, Jonee and yes, we also love the 3-Act Task lessons. We have found them really helpful for presenting the mathematics and in particular, videos can provide visual access that reading a task does not. We have used different ones when considering other topics such as mixing paint.

We have also found that sentence starters support student talk and discussion, and particularly, when teachers craft them to be content-specific, the mathematical talk in the classroom can become really rich. Sentence starters can also be varied for students who have different language strengths and needs and can help students get started in sharing their ideas.  We certainly agree-- these strategies can set up students for success.

Thanks, Bridina. We worked with teachers across urban, rural, small town, and suburban school districts, and we sought to have a variety of classroom sizes as well, while seeking to have at least three or four emergent multilingual students in each classroom. We were trying to develop and test strategies that could be used by mathematics teachers who have a range of language strength and needs on their classrooms. Many mathematics teachers have a wealth of content expertise but may not know how to best integrate language supports into their instruction, while keeping the mathematical content and rigor at grade level.  We found that setting both language and math objectives for lessons and planning ahead for using language strategies facilitated teachers in taking up new instructional strategies.

Thanks so much! We're glad to hear you find these teaching techniques helpful.

Thanks, David! We love seeing students dive into problem solving and talking about the mathematics using strategies like the three reads, sentence starters, and visual representations. And yes--in a related work (see go.edc.org/vam) we have been developing and studying professional learning to support teachers in integrating strategies with their own curriculum. We are also always thinking about how to do more of that. I'd be interested to hear ideas you or others have about resources or tools that might be most helpful to teachers.

Wonderful!  I'll have a look!

Thanks, David, and I also wanted to note that we put in the sentence starter cards that we use during partner work in the sidebar. These prompt student to talk about their diagrams with their partner, but could be used to prompt for other conversations as well; instead of, "In this diagram, I see," it could modified to be about graphs drawn or a problem solving approach. We have found that specifically focusing mathematical talk facilitates student talk more than sentence starters such as, "I agree... " or "I disagree..."

Thanks for pointing out these resources, Kim!